Traveling Waves for Spatially Discrete Systems of FitzHugh--Nagumo Type with Periodic Coefficients

Journal article (2019)

Authors

W.M. Schouten-Straatman Universiteit Leiden
Hermen Jan Hupkes Universiteit Leiden
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Published Date

2019

Language

English

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Abstract

We establish the existence and nonlinear stability of traveling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh--Nagumo system with alternating scale-separated diffusion coefficients. In particular, we view such systems as singular perturbations of spatially homogeneous LDEs, for which stable traveling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multicomponent functional differential equations of mixed type (MFDEs). In order to analyze these equations, we generalize the spectral convergence technique that was developed by Bates, Chen, and Chmaj to analyze the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm properties from the spatially homogeneous to the spatially periodic setting. Our results hence do not require the use of comparison principles or exponential dichotomies.